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This article examines the spatial {dynamics of bed load particles} in water. We focus particularly on the fluctuations of particle activity, which is defined as the number of moving particles per unit bed {length}. Based on a stochastic model recently proposed by \citet{Ancey2013}, we derive the second moment of particle activity analytically; that is the spatial correlation functions of particle activity. From these expressions, we show that large moving particle clusters can develop spatially. Also, we provide evidence that fluctuations of particle activity are scale-dependent. Two characteristic lengths emerge from the model: a saturation length $\ell_{sat}$ describing the length needed for a perturbation in particle activity to relax to the homogeneous solution, and a correlation length $\ell_c$ describing the typical size of moving particle clusters. A dimensionless P\'eclet number can also be defined according to the transport model. Three different experimental data sets are used to test the theoretical results. We show that the stochastic model describes spatial patterns of particle activity well at all scales. In particular, we show that $\ell_c$ and $\ell_{sat}$ may be relatively large compared to typical scales encountered in bed load experiments (grain diameter, water depth, bed form wavelength, flume length...) suggesting that the spatial fluctuations of particle activity have a non-negligible impact on the average transport process.

Steep slope streams show large fluctuations of sediment discharge across several time scales. These fluctuations may be inherent to the internal dynamics of the sediment transport process. A probabilistic framework thus seems appropriate to analyze such a process. In this letter, we present an experimental study of bedload transport over a steep slope flume for small to moderate Shields numbers. The sampling technique allows the acquisition of high-resolution time series of the solid discharge. The resolved time scales range from $10^{-2}$s up to $10^{5}$s. We show that two distinct time scales can be observed in the probability density function for the waiting time between moving particles. We make the point that the separation of time scales is related to collective dynamics. Proper statistics of a Markov process including collective entrainment are derived. The separation of time scales is recovered theoretically for low entrainment rates.

In gravel-bed rivers, bedload transport exhibits considerable variability in time and space. Recently, stochastic bedload transport theories have been developed to address the mechanisms and effects of bedload transport fluctuations. Stochastic models involve parameters such as particle diffusivity, entrainment and deposition rates. The lack of hard information on how these parameters vary with flow conditions is a clear impediment to their application to real-world scenarios. In this paper, we determined the closure equations for the above parameters from laboratory experiments. We focused on shallow supercritical flow on a sloping mobile bed in straight channels, a setting that was representative of flow conditions in mountain rivers. Experiments were run at low sediment transport rates under steady nonuniform flow conditions (i.e., the water discharge was kept constant, but bedforms developed and migrated upstream, making flow nonuniform). Using image processing, we reconstructed particle paths to deduce the particle velocity and its probability distribution, particle diffusivity, and rates of deposition and entrainment. We found that on average, particle acceleration, velocity and deposition rate were responsive to local flow conditions, whereas entrainment rate depended strongly on local bed activity. Particle diffusivity varied linearly with the depth-averaged flow velocity. The empirical probability distribution of particle velocity was well approximated by a Gaussian distribution when all particle positions were considered together. In contrast, the particles located in close vicinity to the bed had exponentially distributed velocities. Our experimental results provide closure equations for stochastic or deterministic bedload transport models., Comment: Submitted to Journal of Geophysical Research